# Solve the trigonometric series equation 1 + sin &#x2061;<!-- ⁡ --> x + s

Solve the trigonometric series equation $1+\mathrm{sin}x+{\mathrm{sin}}^{2}x+{\mathrm{sin}}^{3}x+\cdots =4+2\sqrt{3}$
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Leroy Lowery
$\mathrm{\forall }x\in \mathbb{R}|\mathrm{sin}x|\le 1$ and if $|u|<1$ then $1+u+{u}^{2}+{u}^{3}+...=\frac{1}{1-u}$
$\frac{1}{1-\mathrm{sin}x}=4+2\sqrt{3}$
$\mathrm{sin}x=\frac{\sqrt{3}}{2}$
$x=\left(-1{\right)}^{k}\frac{\pi }{3}+\pi k,k\in \mathbb{Z}$